I highly doubt it.
#1, the 4473s are the actual paper forms you fill out (Fed, state may have their own as well) and they are RETAINED by the FFL DEALER. Until/unless the dealer goes out of business, THEN they are shipped to the ATF.
(there is a time limit the dealer must keep them, 20 years, I think, not certain about that..)
In order for the ATF (or anyone) to check they forms they would have to physically visit the FFL location and check the records. This is done in yearly inspections, or for specific investigations.
#2, since there is, at this time, NO connection AT ALL with the no fly list, other than the misguided ranting of politicians, what govt agency is going to spend its budget sending agents to each and every FFL dealer, with a copy of the NoFly list, to check its 50,000+ names against all the 4473s the dealer has???
Unless I am much mistaken the 4473s are NOT computerized, so any search would have to be "by hand" at each and every dealer in the country. Not only would that entail a truly MASSIVE cost in man hours, I doubt very much if the network of dealers and gun friends in the country would fail to note this, and make it public knowledge.
Now, the background check info, the over the phone data, (which is NOT the 4473, but uses some of the information on it) could be stored on computer, and used to check against every secret list the govt has.
Possibly why that data is required BY LAW to be purged after a set time. It would not surprise me to learn that certain govt agencies are keeping the data, in violation of statute, to use for their own purposes.